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11.2: F Distribution

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    The distribution used for the hypothesis test is a new one. It is called the \(F\) distribution, named after Sir Ronald Fisher, an English statistician. The \(F\) statistic is a ratio (a fraction). There are two sets of degrees of freedom; one for the numerator and one for the denominator.

    For example, if \(F\) follows an \(F\) distribution and the number of degrees of freedom for the numerator is four, and the number of degrees of freedom for the denominator is ten, then \(F \sim F_{4,10}\).

    The \(F\) distribution is derived from the Student's \(t\)-distribution. The values of the \(F\) distribution are squares of the corresponding values of the \(t\)-distribution. One-Way ANOVA expands the \(t\)-test for comparing more than two groups. The scope of that derivation is beyond the level of this course.

    Here are some facts about the \(F\) distribution:

    1. The curve is not symmetrical but skewed to the right.
    2. There is a different curve for each set of \(dfs\).
    3. The \(F\) statistic is greater than or equal to zero.
    4. As the degrees of freedom for the numerator and for the denominator get larger, the curve approximates the normal.
    5. Other uses for the \(F\) distribution include comparing two variances and two-way Analysis of Variance. Two-Way Analysis is beyond the scope of this chapter.
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    Figure \(\PageIndex{1}\)

    References

    1. Data from a fourth grade classroom in 1994 in a private K – 12 school in San Jose, CA.
    2. Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets: Data for Fruitfly Fecundity. London: Chapman & Hall, 1994.
    3. Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets.London: Chapman & Hall, 1994, pg. 50.
    4. Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets. London: Chapman & Hall, 1994, pg. 118.
    5. “MLB Standings – 2012.” Available online at http://espn.go.com/mlb/standings/_/year/2012.
    6. Mackowiak, P. A., Wasserman, S. S., and Levine, M. M. (1992), "A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich," Journal of the American Medical Association, 268, 1578-1580.

    Review

    The graph of the \(F\) distribution is always positive and skewed right, though the shape can be mounded or exponential depending on the combination of numerator and denominator degrees of freedom.

    As in any analysis, graphs of various sorts should be used in conjunction with numerical techniques. Always look of your data!


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